Water is being heated in an electric kettle - Edexcel - A-Level Maths Pure - Question 5 - 2015 - Paper 3
Question 5
Water is being heated in an electric kettle. The temperature, $ heta^ ext{C}$, of the water $t$ seconds after the kettle is switched on, is modelled by the equation
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Worked Solution & Example Answer:Water is being heated in an electric kettle - Edexcel - A-Level Maths Pure - Question 5 - 2015 - Paper 3
Step 1
State the value of $ heta$ when $t = 0$
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Answer
To find the value of heta when t=0, we substitute t=0 into the equation:
heta=120−100e−0=120−100imes1=20.
Thus, the value of heta is 20extC.
Step 2
find the exact value of $ au$, giving your answer in the form $ rac{ ext{ln } a}{b}$, where $a$ and $b$ are integers
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Answer
Given heta=70 when t=40, we can substitute into the equation:
70=120−100e−au.
Rearranging gives:
100e−au=120−70100e−au=50e−au=0.5.
Now, applying natural logarithm:
−au=extln(0.5) au = - ext{ln }(0.5) = ext{ln }(2^{-1}) = -(-1) ext{ln }(2) = rac{ ext{ln } 2}{1}.
Thus, au can be expressed as rac{ ext{ln } 2}{1}.
Step 3
Calculate the value of $T$ to the nearest whole number
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Answer
To find T when heta=100, we use
100=120−100e−au.
Rearranging gives:
100e−au=120−100100e−au=20e^{- au} = rac{20}{100} = 0.2.
Taking the natural logarithm:
−au=extln(0.2)extorau=−extln(0.2).
Now, to find T, we know that au=40, thus:
T=−au=−extln(0.2)extorapproximately93.
Therefore, rounded to the nearest whole number, T is 93.
